A Mathematical Journey to Relativity

This book opens with an axiomatic description of Euclidean and non-Euclidean geometries. Euclidean geometry is the starting point in understanding all other possible geometries and serves as a cornerstone for our basic intuition for vector spaces. The generalization to non-Euclidean geometry is the following step to develop the language for Special and General Relativity. This book discusses Special and General Relativity starting from a full geometric point of view. Differential geometry is presented in the simplest possible way and it is applied in describing the physical world. The final result of this construction is deriving the Einstein field equations to describe gravitation and spacetime. Possible solutions, and their physical implications are also discussed: the Schwarzschild metric, the relativistic trajectory of planets, the deflection of light, the black holes, cosmological solutions like de Sitter, Friedmann-Lemaitre-Robertson-Walker, and Goedel ones, the dark energy problem. This clearly written, self-contained book includes details of all proofs, and provides solutions to all the problems or tips on solving them. This book is designed for undergraduate students and for all readers who want a first geometric approach to Special and General Relativity.